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A simplified human birth model:
Translation of a Rigid Cylinder
Through a Passive Elastic Tube
Roseanna Gossmann1 , Alexa Baumer2 , Lisa Fauci1 ,
Megan C. Leftwich2
Tulane University1 , George Washington University2
Work supported in part by NSF DMS 1043626
Motivation
Vaginal delivery is linked to
I shorter post-birth hospital stays
I lower likelihood of intensive care
stays
I lower mortality rates [1]
Fluid mechanics greatly informs the
total mechanics of birth.
I vernix caseosa
I amniotic fluid
[1] C. S. Buhimschi, I. A. Buhimschi (2006). Advantages of vaginal delivery, Clinical obstetrics and gynecology.
Fig. 1: “HumanNewborn” by Ernest F - Own work. Licensed under CC BY-SA 3.0 via Commons - https://
commons.wikimedia.org/wiki/File:HumanNewborn.JPG#/media/File:HumanNewborn.JPG
Fig. 2: “Postpartum baby2” by Tom Adriaenssen - http://www.flickr.com/photos/inferis/110652572/.
Licensed under CC BY-SA 2.0 via Commons - https://commons.wikimedia.org/wiki/File:Postpartum_baby2.
jpg#/media/File:Postpartum_baby2.jpg
Physical Experiment
Materials:
I
Rigid acrylic cylinder (fetus)
I
Passive elastic latex tube
(birth canal)
I
Viscous fluid - methyl
cellulose and water
(amniotic fluid)
Physical Experiment
The rigid cylinder is pulled at a constant velocity through the
center of the elastic tube. The entire system is immersed in fluid.
The Model: Solid Behavior
Elastic Tube
I Tube modeled by network of Hookean springs.
I Force at xl due to spring from xm :
f(xl ) = τ
kxm − xl k
−1
∆lm
(xm − xl )
kxm − xl k
where τ =constant, ∆lm =resting distance between points
circumference
z
The Model: Solid Behavior
I Spring constant τ chosen to match elastic properties to physical
experiment by setting the total elastic energy of the discrete spring
system [2] equal to the stored energy in the continuous elastic tube used
in the physical experiment:
X
springs
τ
1
(||xm − xl || − ∆lm )2 = Aβ 2 L
2∆lm
2
where ∆lm =resting distance between points xm , xl , A = EI , E =Young’s
modulus of the fluid, I =tube’s second moment of area, L =tube length.
[2] H. Nguyen and L. Fauci (2014). Hydrodynamics of diatom chains and semiflexible fibres, J. R. Soc. Interface.
The Model: Solid Behavior
Rigid Inner Rod
I A constant velocity u is specified in the z-direction.
Figure: Discretization of rigid cylinder and elastic tube position in fluid at
beginning of simulation.
The Model: Fluid Dynamics
Fluid Behavior is governed by the Stokes equations:
0 = −∇p + µ∆u + f,
0=∇·u
For a forces at discrete points xk in three dimensions:
0 = −∇p + µ∆u +
K
X
f0 δ(x − xk ), ∇ · u = 0
k=0
=⇒ u(x) =
p(x) =
K (fk · x)x
1 X 1
fk +
,
8πµ
rk
rk3
k=0
K
X
fk · ∇G (r )
k=0
where rk = ||xk − x||2 , ∆G (r ) = δ(r ).
The Model: Fluid Dynamics
The method of regularized stokeslets [3],[4] uses a continuous function φε to
approximate the dirac-δ function, approximating the Stokes equations for K
point forces by
P
0 = −∇p + µ∆u + Kk=0 f0 φε (x − xk ), ∇ · u = 0,
which has solution
u(x) =
K
1X
[(fk · ∇)∇Bε (|x − xk |) − fk Gε (|x − xk |)] ,
µ
k=1
p(x) =
K
X
[fk · ∇Gε (|x − xk |)] ,
k=1
where ∆Bε = Gε , ∆Gε = φε (r ) =
15ε4
.
8π(r 2 +ε2 )(7/2)
Here, µ is viscosity, xk are points on discretized tube and rod, fk is the force at
that point, and ε is a regularization parameter.
[3] R. Cortez (2001). Method of Regularized Stokeslets, SIAM Journal of Scientific Computing.
[4] R. Cortez, L. Fauci, A. Medovikov (2005). The method of regularized Stokeslets in three dimensions: analysis,
validation, and application to helical swimming, Physics of Fluids.
The Model: Numerical Solution
Using the solution to the regularized Stokes equations for a given blob function, we can
(1) find the velocity induced on the rod by spring forces in the tube,
(2) solve for any additional forces on the rod necessary to achieve its
prescribed velocity,
(3) evaluate the velocity and pressure at every point in the system,
(4) update the tube and rod positions using these velocities one step forward
in time.
Choosing a Regularization Parameter: Effect of ε
∆ s = tubelength/10 = 0.343, cylinder = 1.5tube
50
eps = 0.5∆ s
eps = 1.0∆ s
eps = 1.5∆ s
45
40
Force, in kg.cm/s 2
35
30
25
20
15
10
5
0
0
20
40
60
80
100
120
140
Time, in s
Figure: Computed drag force as a function of time for a rigid cylinder moving
through a rigid tube. Force varies greatly with ε.
Drag Force on a Cylinder
Slender body theory provides an asymptotic approximation to the drag force on
a long, slender cylinder moving along its axis through a viscous fluid [5]:
F=−
4πµUL
2 ln(L/r ) − 1
This value can be compared to numerically approximated drag force on a cylinder of fixed radius and varying length, for different values of the regularization
parameter ε.
[5] C. Pozrikidis (2011). Introduction to theoretical and computational fluid dynamics, Oxford University Press.
Force Error
102
Relative error in drag force on cylinder
log(relative drag force error)
101
100
10-1
L/r = 10
L/r = 20
L/r = 30
L/r = 40
L/r = 50
L/r = 60
L/r = 80
10-2
10-3
10-3
10-2
10-1
100
101
log(epsilon/∆ s)
Figure: Error in drag force computation is minimized for ε between 0.07∆s
and 0.08∆s, where ∆s is the approximate mesh size of the discretization.
Velocity Error – Leak Test
Velocity error in leak test
maximum velocity error
1
L/r = 10
L/r = 20
L/r = 30
L/r = 40
L/r = 50
L/r = 60
L/r = 80
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
epsilon/ ∆ s
Figure: Measuring the velocity at points on the surface of the cylinder in
between the points at which the velocity is specified supplies a measure for
how closely the discrete cylinder approximates a solid object. Error decreases
as the size of ε increases.
Velocity Profile Between Cylinder and Solid Tube
0.8
Velocity profile between cylinder and solid tube
0.7
0.6
velocity
0.5
0.4
0.3
0.2
0.1
0
1.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
radial position, r
Figure: The velocity profile between an infinitely long rigid cylinder of radius
t )−ln(r ))
. This is
Rc and an infinitely long rigid tube of radius Rt is u(r ) = U(ln(R
ln(Rt )−ln(Rc )
compared with the velocity computed using the method of regularized
stokeslets with varying values for ε.
Velocity Error
Maximum error in velocity profile between cylinder and tube
0.5
0.45
maximum velocity error
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
epsilon/ ∆ s
Figure: Error in the velocity profile is minimized for ε ≈ 0.3∆s.
A Simulation
I
Tube radius: 1.613 cm
I
Cylinder radius: 1.27 cm
I
Tube length: 10.8 cm
I
Cylinder length: 13.2 cm
I
Fluid viscosity: 0.0305 Pa·s
I
Elastic Young’s modulus: 0.01GPa
I
Regularization parameter: ε = 0.5∆s
System Behavior
Fluid Velocity
Fluid Pressure
Tube Buckling
Pulling Force
Total pulling force on rigid cylinder
3.5
3
drag force (N)
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
time (s)
30
35
40
45
50
Future Work
I
Further analysis of tube buckling behavior
I
I
I
I
I
Different diameter and length of rigid cylinder
Different velocity of rigid cylinder
Different elasticity of tube
Find the “perfect” regularization parameter value
Increase realism
I
I
active elastic tube / modeling peristalsis
more accurate geometry
Slides available at
math.tulane.edu/~rpealate